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, Volume 118, Issue 2, pp 135–149 | Cite as

A proof of Selberg's orthogonality for automorphic L-functions



Let π and π′ be automorphic irreducible cuspidal representations of GL m (Q A ) and GL m (Q A ), respectively. Assume that π and π′ are unitary and at least one of them is self-contragredient. In this article we will give an unconditional proof of an orthogonality for π and π′, weighted by the von Mangoldt function Λ(n) and 1−n/x. We then remove the weighting factor 1−n/x and prove the Selberg orthogonality conjecture for automorphic L-functions L(s,π) and L(s,π′), unconditionally for m≤4 and m′≤4, and under the Hypothesis H of Rudnick and Sarnak [20] in other cases. This proof of Selberg's orthogonality removes such an assumption in the computation of superposition distribution of normalized nontrivial zeros of distinct automorphic L-functions by Liu and Ye [12].


Automorphic L-function Selberg's orthogonality 

Mathematics Subject Classification (2000)

11F70 11N05 11F66 11M26 11M41 


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© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Department of MathematicsShandong UniversityJinanChina
  2. 2.Department of MathematicsCapital Normal UniversityBeijingChina
  3. 3.Department of MathematicsThe University of IowaIowa CityUSA

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